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A very tall vertical cylinder is filled ...

A very tall vertical cylinder is filled with a gas of molar mass M under isothermal conditions temperature T. the density and pressure of the gas at the base of the container is `rho_(0)` and `P_(0)`, respectively
Select the correct statement

A

The density of gas cannot be uniform throughout the cylinder

B

The density of gas cannot be uniform throughout the cylinder under isothermal conditions

C

The rate of change of density `|(drho)/(dh)|=(rhoMg)/(RT)`

D

All of the above

Text Solution

Verified by Experts

The correct Answer is:
D
Considering the equilibrium of differential layer of a thickness dh and height h, we have
`(P+dP)A+dmg=PA`
or `A dP=-g dm=0grhoA dh`
From ideal gas equation, we have
`P=(rhoRT)/(M)` or `rho=(PM)/(RT)`
From eqns. (1) and (2), we get
`dp=-(PM)/(RT)g dh`
or `In (P)/(P_(0))=-(Mg)/(RT)h`
or `P=P_(0)e^(-Mgh//RT)`
In a similar manner if we eliminate P with the of ideal gas equation, we get
`dP=(drhoRT)/(M)`
The eqn. becomes
`drho(RT)/(M)=-rhog dh`
or `int_(P_(0))^(rho) (drho)/(rho)=-(Mg)/(RT)int_(0)^(h)dh`
or `rho=rho_(0)e^(-Mgh//RT)`
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